Gen. Math. Notes, Vol. 15, No. 1, March, 2013, pp. 56-71
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Periodic Solution of Nonlinear System of
Integro-Differential Equations Depending on
the Gamma Distribution
Raad N. Butris
Assistant Professor, Department of Mathematics
Faculty of Science, University of Zakho
E-mail: raadkhlka_khlka@yahoo.com
(Received: 23-10-12 / Accepted: 18-12-12)
Abstract
In this paper we investigate the periodic solution of nonlinear system of
integro- differential equations depending on the gamma distribution by using the
numerical analytic method for investigate periodic solutions of ordinary
differential equation which given by Samoilenko A. M. .These investigations lend
us to the improving and extending the above method. Also we expand the results
gained by Butris R. N. to change the periodic system of nonlinear integrodifferential equations to periodic system of nonlinear integro-differential
equations depending the on gamma distribution.
Keywords: Numerical-analytic methods existence of periodic solutions,
nonlinear integro-differential equations, subjected, depending on the gamma
distribution.
Periodic Solution of Nonlinear System of…
1
57
Introduction
They are many subjects in physics and technology using mathematical methods
that depends on the linear and nonlinear integro-differential equations, and it
become clear that the existence of periodic solutions and it is algorithm structure
from more important problems, to present time where many of studies and
researches [3, 4, 5, 8, 9] dedicates for treatment the autonomous and nonautonomous periodic systems and specially with integro-differential equations.
Numerical-analytic method [2,3,4,6,8] owing to the great possibilities of
exploiting computers are becoming versatile means of the finding and
approximate construction of periodic solutions of integro-differential equations.
Samoilenko [7] assumes the numerical-analytic method to study the periodic
solutions for ordinary differential equations and it is algorithm structure and this
method include uniformly sequences of periodic functions and the results of that
study is using of the periodic solutions on wide range in the difference of new
processes industry and technology as in the studies[2,6,8,9].
Butris R. N. [1] has been used the numerical-analytic method of periodic solution
for ordinary differential equations which were introduced by Samoilenko A. M.
[7] to study the periodic solution of the system, nonlinear integro-differential
equation which has the form:
= , , , ⋯ 1.1
Where ∈ ⊂ , is a closed and bounded domain. The vectors functions
, , and , are continuous functions in , , and periodic in of period
.
In this study we have employed the numerical-analytic method of Samoilenko
A.M. [7] to investigate the existence and approximation of periodic solution for
nonlinear system of integro-differential equations which depending on the
gamma distribution. The study of such integro-differential equations leads to
improving and extending Samoilenko A.M. [7] method.
Thus, the integro-differential equtions which depending on the gamma
distribution that we have introduced in this study, becomes more general and
detailed than those introduced by Butris R. N. [1]. The study is considered a
theoretical one, however, the results that we have got, may several applications in
field physical as well as mathematical problems.
58
Raad. N. Butris
In this paper, we consider the periodic integro-differential equations which
depending on the gamma distribution:
= , , , , ⋯ 1.2
, , , where ∈ ⊂ , is a closed bounded domain.
The vector functions , , , and , , , are defined on the
domain:
, , , ∈ ! × #0, % × × !
= −∞, ∞ × #0, % × × !
⋯ 1.3
Continuous in the totality of variables periodic in of period and satisfies the
inequalities:
, |, , , |
|, , , , | ≤ ++,
≤ ++,
⋯ 1.4
|, , , ! , ! − , , , . , . | ≤ |+, ||! − . | + 0|! − . |
⋯ 1.5
|, , , ! − , , , . |
≤ 2|+, ||! − . |
⋯ 1.6
for all ∈ ! and , ! , . ∈ , where + = +! , +. , ⋯ , + is a positive constant
vectors and gamma distribution is defined by the following equation:
, =
4
where
,5! 6 5
, > 0 ,C
A
Γ
Γ + 1 > B
? . A
<
+ 1
@
C
2
1.8
, 6 5 B ⋯
= − ++,
2
Γ @
⋯ 1.7
We define the non-empty sets as follows:
4
EG
EF = − ++,
Furthermore, we suppose that the greatest eigen value IJKL of the matrix
Periodic Solution of Nonlinear System of…
59
Λ = #+, N + 02 % does not exceed unity, i.e.
.
IJKL Λ < 1
⋯ 1.9
Lemma 1.1 [7]: Let be a continuous vector function defined on the interval
#0, %, then
1
P − P ≤ max ||
∈#Q,%
Q
Q
Where = 21 − . (For the proof see [7]).
By using Lemma 1.1, we can state and prove the following Lemma.
Lemma 1.2: Suppose that the function of gamma distribution , is continuous
on the interval #0, %. Then
1
P, − , P ≤ +, Q
Q
is hold for all values of .
Proof: Taking
1
P, − , P ≤ 1 − |I, | + |I, | =
Q
Q
So that
Q
,5! 6 5
#1 − + − % =
Γ
= +,
1
P, − , P ≤ +, ⋯ 1.10
Q
Q
,5! 6 5
,5! 6 5
= 1 − + ≤
Γ
Γ
≤
For all ∈ #0, % and ≤ . .
60
Raad. N. Butris
2
Approximate Solution
The investigation of periodic approximate solution of the problem (1.2) is
formulated by the following theorem.
Theorem 2.1: If the problem (1.2) satisfy the inequalities (1.4),(1.5),(1.6) and the
conditions (1.7), (1.8) has a periodic solution = , , , Q , then the
sequence of functions
V
J! , , , Q = Q + #, , , J , , , Q , U, U, ,
Q
, J U, U, , Q U −
VW
V
1
, , , J , , , Q ,
Q
, U, U, , J U, U, , Q U% ⋯ 2.1
V
With
Q , , , Q = Q ,
X = 0,1,2, ⋯
is periodic in of period , and uniformly convergent as X → ∞ in the domain
, , , Q ∈ ! × #0, % × EF ⋯ 2.2
To the limit function Q , , , Q which is defined on the domain (2.2),
periodic in of period and satisfying the system of integral equations.
V
, , , Q = Q + #, , , , , , Q , U, U, ,
Q
, U, U, , Q U −
VW
V
1
, , , , , , Q ,
Q
, U, U, , U, U, , Q U% ⋯ 2.3
V
Which is a unique solution of the problem (1.2) provided that:
| Q , , , Q − Q | ≤ + +, ⋯ 2.4
And
| Q , , , Q − J , , , Q | ≤ ΛJ Z − Λ5! + +, ⋯ 2.5
for all X ≥ 1 and ∈ ! , where Z is the identity matrix.
Periodic Solution of Nonlinear System of…
61
Proof: Consider the sequence of functions ! , , , Q , . , , , Q , ⋯,
J , , , Q , ⋯ defined be recurrence relation (2.1). Each of these sequence
of functions is defined and continuous on the domain (1.3) and periodic in of
period .
Now, by Lemma 1.1, 1.2 and using (2.1), when X = 0, we get:
|! , , , Q − Q |
V
≤ 1 − |, , , Q 4, U, U, , Q 4U| +
Q
V
V
+ |, , , Q 4, U, U, , Q 4U| ≤
V
≤ 1 − + +, + + +, Q
= + +, #1 − + − %
= 21 − + +,
= + +,
So that
|! , , , Q − Q | ≤ + +, ⋯ 2.6
i.e. ! , , , Q ∈ , for all ∈ ! , Q ∈ EF .
Also from (2.6), we have:
|! , , , Q − Q , , , Q | = P , , , ! , , , Q −4
4, , , , Q P ≤
≤ 2 |, ||! , , , Q − Q | ≤
62
Raad. N. Butris
,5! 6 5
≤ 2
+ +, Γ
, 6 5
=2
+ +,
2
Γ
, 6 5
|! , , , Q − Q , , , Q | ≤ 2
+ +, ⋯ 2.7
Γ
2
So that
i.e. ! , , , Q ∈ ! , for all ∈ ! and Q ∈ EF .
Thus by mathematical induction, we find that:
|J , , , Q − Q | ≤ + +, ⋯ 2.8
For all ∈ ! and Q ∈ EF .
i.e. J , , , Q ∈ , for all ∈ ! and Q ∈ EF .
And from (2.8), we get:
|J , , , Q − Q , , , Q | ≤ + +, 2
For all ∈ ! and Q ∈ EF .
, 6 5 Γ 2
i.e. J , , , Q ∈ ! , for all ∈ ! and Q ∈ EF .
Where J , , , Q = \
X = 0,1,2, ⋯ .
, , , J , , , Q ,
We claim that the sequence of functions (2.1) is uniformly convergent on the
domain (2.2).
By using the Lemmas 1.1, 1.2 and putting X = 1 in (2.1), we have:
|. , , , Q − ! , , , Q | = |Q 4 + #, , , ! , , , Q ,
V
Q
1
, U, U, , ! U, U, , Q U − , , , ! , , , Q ,
V
Q
Periodic Solution of Nonlinear System of…
63
V
, U, U, , ! U, U, , Q U % − Q − #, , , Q ,
V
V
, U, U, , Q U −
V
V
Q
1
, , , Q , U, U, , Q U 4%| ≤
Q
V
≤ 1 − |, |N + 02|! , , , Q − Q | +
Q
+ |, | N + 02|! , , , Q − Q | ≤
≤ +, + +,
N + 02
2
|. , , , Q − ! , , , Q | ≤ + +, . N + 02 ⋯ 2.9
2
|J , , , Q − J5! , , , Q | ≤ + +, J # N + 02%J5! 2
⋯ 2.10
For all X ≥ 1.
Suppose that the following inequality is true
Now, we shall prove the following:
|J! , , , Q − J , , , Q | ≤
≤ 1 − +, N + 02|J , , , Q − J5! , , , Q | +
Q
+ +, N + 02|J , , , Q − J5! , , , Q | ≤
J5!
≤ 1 − +, N + 02+ +, J ]N + 02 ^
+
2
Q
64
Raad. N. Butris
J5!
J
+ +, N + 02+ +, ]N + 02 ^
=
2
J5!
= +,
]N + 02 ^
+ =
2
2
J
= + +, J! ]N + 02 ^ =
2
J!
J
= + +, ]+, N + 02 ^ =
2
And hence
|J! , , , Q − J , , , Q |
J
≤ ++, ]+, N + 02 ^ ⋯ 2.11
2
For all X ≥ 0 .
From (2.11) we conclude that for any_ ≥ 1, we have the inequality
`5!
|J` , , , Q − J , , , Q | ≤ a ΛJb + +, Such that
|J` , , , Q − J , , , Q | ≤
bcQ
d
≤ a|J!b , , , Q − Jb , , , Q | ≤
bcQ
d
≤ a ++, ΛJ!b =
bcQ
d
≤ + +, Λ a Λb! =
J
bcQ
≤ + +, ΛJ E − Λ5! =
So that
Periodic Solution of Nonlinear System of…
65
|J` , , , Q − J , , , Q |
⋯ 2.12
≤ ΛJ E − Λ5! + +, for all _ ≥ 1, where Z is identity matrix.
from (2.12) and the condition (1.9), we find that:
lim ΛJ
J→d
⋯ 2.13
=0
Relations (2.12) and (2.13) prove the uniform convergence of the sequence of
functions (2.1) on the domain (2.2).
Let
lim J , , , Q = Q , , , Q ⋯ 2.14
J→d
Since the sequence of functions (2.2) is periodic in of period , then the limiting
function Q , , , Q is also periodic in of period .
Moreover, by Lemmas 1.1, 1.2 and inequality (2.12) the inequalities (2.4) and
(2.5) are holds.
Finally, we have to show that , , , Q is a unique solution of the system
(1.1).Assume that h, , , Q is another solution of the system (1.1),
i.e.
h, , , Q = Q + #, , , h, , , Q , Q
V
V
U, U, ,
1 , hU, U, , Q U − , , , h, , , Q ,
Q
,
V
VW
U, U, , hU, U, , Q U% ⋯ 2.15
Now, we prove that , , , Q = h, , , Q for all Q ∈ EF and to
do this, we need to drive the following inequality:
|h, , , Q − , , , Q | ≤ ΛJ E − Λ5! +∗ +, where+∗ = Xj |, h, , , Q , , , , Q 4,
Lk ∈lmn
⋯ 2.16
66
Raad. N. Butris
, , , , , , , Q 4|
Suppose that (2.15) is true for X = _, i.e.
|h, , , Q − , , , Q | ≤ Λ` E − Λ5! +∗ +, Then
|h, , , Q − , , , Q | ≤
≤ 1 − +, N + 02|h, , , Q − , , , Q | +
Q
+ +, N + 02|h, , , Q − , , , Q | ≤
≤ 1 − +, N + 02Λ` Z − Λ5! +∗ +, +
Q
+ +, N + 02Λ` Z − Λ5! +∗ +, =
= Λ`! Z − Λ5! +∗ +, By induction, inequality (2.16) is true for X = 0,1,2, ⋯.
Thus from (2.14) and (2.16), we have:
lim |h, , , Q − J , , , Q | = 0
J→d
And hence
lim J , , , Q = h, , , Q J→d
By the relation (2.14), we get:
, , , Q = h, , , Q i.e. , , , Q is a unique solution of (1.1) on the domain (1.2). ∎
Periodic Solution of Nonlinear System of…
3
67
Existence of Solution
The problem of existence of a periodic solution of period of the system (1.1) is
uniquely connected with the existence of zeros of the function Δ0, 0, , Q which has the form:
1
Δ0, 0, , Q = #, , , Q , , , Q ,
Q
, , , , Q , , , Q %
⋯ 3.1
Where Q , , , Q is the limiting function of the sequence of functions
(2.1).
The function (3.1) can find only approximately, say by computing the following
functions:
1 Δq 0, 0, , Q = #, , , J , , , Q ,
Q
, , , , J , , , Q %
and X = 0,1,2, ⋯.
Now, we prove the following theorem.
⋯ 3.2
Theorem3.1: Let all assumptions and conditions of theorem 2.1 were given. Then
the inequality:
|r0, 0, , Q −rJ 0, 0, , Q | ≤ sJ! Z − s5! + +, ⋯ 3.3
will be satisfied for all X ≥ 0, Q ∈ EF .
Proof: By the relations (3.1) and (3.2), the estimate
|Δ0, 0, , Q −Δq 0, 0, , Q | ≤
1
≤ |, | N + 02| Q , , , Q − J , , , Q | ≤
Q
1
≤ +, N + 02ΛJ Z − Λ5! + +, =
Q
68
Raad. N. Butris
= ΛJ! Z − Λ5! + +,
Thus the inequality (3.2) is hold for all X ≥ 0 .
Next, we prove the following theorem taking into account at the inequality (3.3)
will be satisfied for all X ≥ 0 .
Theorem 3.2: If the system (1.1) satisfies the following condition
(i)The sequence of functions (3.2) has an isolated singular point Q = Q ,
rJ 0, 0, , Q ≡ 0, for all ∈ ! .
(ii)The index of this point is nonzero;
(iii)There exists a closed convex domain E∗ belonging to domain EF and
possessing a unique singular point Q such that on it is boundary ulm∗ the
following inequality is holds
vw‖rJ , , , Q ‖ ≤ ‖sJ Z − s5! + +, ‖ ⋯
3.4
when ∈ ulm∗ for all X ≥ 0 . Then the system (1.1) has a periodic solution
= , , , Q for which 0, 0, , Q belongs to the domain E∗ .
Proof: By using the inequality (3.1) we can prove the theorem 7.1[7].
Remark 3.1. [7]: When = ! , i.e. when Q is a scalar, the existence of
solution can be strengthens by giving up the requirement that the singular point
shout be isolated, thus we have
Theorem3.3: Let the system of nonlinear integro-differential equations (1.1) are
defined on the interval #j, y%. Suppose that for X ≥ 0, the function
rJ 0, 0, , Q defined according to formula (3.2) satisfies the inequalities:
min
K{|L
k |}5{
4
max
‖Δq , , , Q ‖ ≤ −~J
K{|Lk |}5{
‖Δq , , , Q ‖ ≥ ~J
;
.
€
⋯ 3.5
Then the system (1.1) has a periodic solution = , , Q for which
Q ∈ #j + ℎ, y − ℎ%, where ℎ = ‖+ +, ‖ . and ~J = ‖ΛJ! Z − Λ5! + +, ‖ .
Proof: Let ! and . be any two points on the interval #j, y% such that:
4
Δq 0, 0, , ! =
Δq 0, 0, , . =
min
K{|Lk |}5{
max
K{|Lk |}5{
Δq 0, 0, , Q ;
Δq 0, 0, , Q .
€
⋯ 3.6
Periodic Solution of Nonlinear System of…
69
Taking into account inequalities (3.3) and (3.5), we have
4 Δ0, 0, , ! = Δq 0, 0, , ! + #Δ0, 0, , ! − Δq 0, 0, , ! % ‚
Δ0, 0, , . = Δq 0, 0, , . + #Δ0, 0, , . − Δq 0, 0, , . %
⋯ 3.7
It follows from the inequalities (3.7) and the continuity of the function
Δ0, 0, , Q , that there exist an isolated singular point Q , Q ∈ #! , . %, such
that Δ0, 0, , Q ≡ 0, this means that the system (1.1) has a periodic solution
= , , , Q for which Q ∈ #j + ℎ, y − ℎ%. ∎.
Theorem3.4: If the function r0, 0, , Q is defined by
Δ: EF → ,
1 Δ0, 0, , Q = #, , , Q , , , Q ,
Q
, , , , Q , , , Q %
⋯ 3.7
Where Q , , , Q is a limit of the sequence of functions (2.1). Then the
following inequalities are holds
|Δ0, 0, , Q | ≤ M M… ⋯ 3.8
And
|Δ0, 0, , Q! − Δ0, 0, , Q . | ≤
For all Q , Q! , Q . ∈ EF .
2
ΛZ − Λ5! +, ⋯ 3.9
Proof: From the properties function Q , , , Q established theorem 2.1; it
follows that function Δ0, 0, , Q is continuous and bounded by + +, .
By using (3.7), we get:
1
|Δ0, 0, , Q! − Δ0, 0, , Q . | = † 4 #, , , Q , , , Q! ,
Q
1
, , , , Q , , , Q! % − #, , , Q , , , Q . ,
, , , , Q , , , Q . % ≤
Q
70
Raad. N. Butris
1
≤ +, N + 02| Q , , , Q! − Q , , , Q . | ≤
Q
2
≤ +, N + 02 . | Q , , , Q! − Q , , , Q . | =
2 =
2
Λ| Q , , , Q! − Q , , , Q . |
And hence
|Δ0, 0, , Q! − Δ0, 0, , Q . | ≤
≤
2
Λ | Q , , , Q! − Q , , , Q . |+, ⋯ 3.10
Where Q! , , , Q and Q . , , , Q are the solutions of the integral
equation:
, , , Q ` = Q ` + #, , , , , , Q ` , Q
, U, U, , Q ` U −
,
With
Q ` , , , Q = Q ` ,
V
VW
V
V
U, U, ,
1 , , , , , , Q ` ,
Q
U, U, , U, U, , Q U% ⋯ 3.11
_ = 1,2.
From (3.11), we have:
| Q , , , Q! − Q , , , Q . | ≤ |Q! − Q . | +
+1 − +, N + 02| Q , , , Q! − Q , , , Q . | +
Q
+ +, N + 02| Q , , , Q! − Q , , , Q . | ≤
≤ |Q! − Q . | + +, N + 02| Q , , , Q! − Q , , , Q . | ≤
≤ |Q! − Q . | + Λ| Q , , , Q! − Q , , , Q . |
Thus
Periodic Solution of Nonlinear System of…
71
| Q , , , Q! − Q , , , Q . | ≤ Z − Λ5! |Q! − Q . | ⋯ 3.12
Using the inequality (3.12) in (3.10), we get (3.9).
Remark 3.2. [4]: The theorem 3.4 ensure the stability solution of the system (1.1)
when there is a slight change on the point Q accompanied with noticeable change
in the function Δ0, 0, , Q .
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